Linear Algebra Section 4.4 Flashcards

Linear Combinations

• A vector v in a vector space V is a linear combination of vectors u1, u2, ..., uk if it can be expressed as c1u1 + c2u2 + ... + ckuk, where c1, c2, ..., ck are scalars.

Spanning Sets

• A subset S = {v1, v2, ..., vk} of a vector space V is a spanning set if every vector in V can be expressed as a linear combination of vectors in S.
• If S spans V, it means that the entirety of V can be reached by taking linear combinations of vectors in S.

Span of a Set

• The span of a set of vectors S = {v1, v2, ..., vk} is defined as the collection of all linear combinations of the vectors in S.
• Denoted as span(S) or span{v1, v2, ..., vk}, it represents all possible linear combinations using real numbers as coefficients.
• If span(S) = V, then it is stated that V is spanned by S.

Subspace Properties

• Span(S) is a subspace of the vector space V that includes all linear combinations of vectors in S.
• It is the smallest subspace containing S, meaning any other subspace that contains S must also include span(S).

Linear Dependence and Independence

• A set of vectors S = {v1, v2, ..., vk} is linearly independent if the equation c1v1 + c2v2 + ... + ckvk = 0 only holds for the trivial solution c1 = 0, c2 = 0, ..., ck = 0.
• If there are nontrivial solutions (other than the trivial one), then the set S is linearly dependent.

Testing Dependence/Independence

• To test whether a set S is linearly independent or dependent:
  • Set up the equation c1v1 + c2v2 + ... + ckvk = 0 and transform it into a system of linear equations.
  • Apply Gaussian elimination to assess the uniqueness of the solution.
  • If the solution is unique (only trivial), S is independent; if there are nontrivial solutions, it is dependent.

Theorem on Linear Independence

• A set of vectors S = {v1, v2, ..., vk}, with k ≥ 2, is linearly independent only if at least one vector can be expressed as a linear combination of the others.

Corollary on Two Vectors

• Two vectors u and v in a vector space V are linearly independent if and only if neither is a scalar multiple of the other.